What can I do to move a file to a new folder and leave its import image as plane intact?
Alright, I found it now. I clicked on File > External Data > Find Missing files, and the File Browser opened, and I clicked the 'Find Missing Files' button on the top right. The folder open in the File Browser needs to have the image in it or it won't work. If you aren't sure where it is, you can click the top entry in the System tab at left and have it search the whole hard disk, or click the up arrow at the top of the folder and file list in the main window area until you are confident it must be somewhere in there.

The image was tinted deep magenta after I did this, but it was there. Closing the file and reopening it caused that to disappear, it was an artifact.